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De Moivre’s Theorem

  1. If n is any rational number, then (cos θ + i sin θ)n = cos = i sin.
  2. If z = (cos θ1 + i sin θ1) (cos θ2 + i sin θ2) (cos θ3 + i sin θ3) … (cos θn + i sin θn) then z = cos (θ1 + θ2 + θ3 + … + θn) + i sin (θ1 + θ2 + θ3 + … + θn), where θ1, θ2, θ3θn R.
  3. If z = r(cos θ + i sin θ) and n is a positive integer, then z1/n = r1/n 59863.png
     
    where k = 0, 1, 2, 3, …, (n – 1).
     
    Deductions: If n Q, then
    1. (cos θ i sin θ)n = cos i sin
    2. (cos θ + i sin θ)n = cos i sin
    3. (cos θ i sin θ)n = cos + i sin
    4. (sin θ + i cos θ)n
       
      = cos n 59857.png
This theorem is not valid when n is not a rational number or the complex number is not in the form of cos θ + isin θ.

The nth root of unity

Let x be the nth root of unity. Then
xn = 1
= 1 + i(0)
= cos 0° + i sin 0°
= cos (2kπ + 0) + i sin (2kπ + 0)
= cos 2kπ + i sin 2kπ (where k is an integer)
 
x = cos 59851.png k = 0, 1, 2, …, n – 1
 
Let α = cos (2π/n) + i sin (2π/n). Then the nth roots of unity are αt (t = 0, 1, 2, …, n – 1), i.e., the nth roots of unity are 1, α, α2, …, αn – 1.




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